Intro Question: Eight Queens
useful ... Pythagorean Triples
3, 4, 5 is the most common one ... but there are an infinite amount of them
5, 12, 13
here is a link to them: https://www.wikiwand.com/en/Pythagorean_triple (easiest reading)
Primitive means they can't all be divided by n.
we can make as many as we want ... check out this graphic:
we notice a pattern between the primitives and the non-primitives
multiplied by value k ... but the angles shouldn't change at all.
the ratio remains the same, which means our movement remains the same.
this is true of ALL right triangles, whether or not they are triples.
let's try another right triangle, with legs of size one.
what's the hypotenuse?
let's walk through a proof together.
This is going to be AWESOME. Your brain will explode with joy.
what about with legs of size Two? Three?
idea of simplifying radicals goes here ... best way to explain it is laundry.
Switch place and time. We are now geocentric.
this creates an interesting idea--the shell, with stars as the end of the universe
1 Celestial Unit is key here
eventually, the sine is the height of the star, and the cosine is how far away the star is on the horizontal plane.
we've created the unit circle. Which I've adapted here: https://www.desmos.com/calculator/fyzainx7b5
Some of these Triangles are "easy" to find some values for:
45 - 45 - 90
30 - 60 - 90
But all triangles have these relationships between size and angles ... and if we can find the ratio, we can find the other sides.
THIS ALLOWS US TO SOLVE THE TRIANGLE.
we have tables with these solved out ratios for us (let us NASA because we can):
as the ratio values are consistent, so too is our ability to find the values, the heights ... or the angle.
This is what our calculators do, and it's awesome.
So let's practice with some triangles. For our first round, let's assume that the angle C is a right angle. For these problems, A or B are angles, a, b, c are sides:
A = 30, c = 10
A = 21, b = 30
a = 5, b = 12
a = 10, c = 17
A = 25, B = 65
Word Problem #1: How tall is the Bennington Monument? We measure the angle we made from us to the monument--it's 15 degrees. On flat land, even with the monument, we walk 75 feet towards the monument, then measure the elevation again. Angle of elevation is now 18.5 degrees.
Word Problem #2: There's a Fire! Fire towers are set up with known distances between them. For a reason. Two fire towers are set up 30 km apart. Degrees used ... 54 degrees and 61 degrees ... how far away is the fire?
The second one is interesting ... there seems to be a little more going on here than meets the eye.
Creation Number One: The Law of Sines